Projective to Affine?

Question

Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help to look at the projective closures (assuming I don't know of any other method for proving they are isomorphic)? How does one go back to the affine case from the projective closure (or "projectivization")? Sorry if this sounds confusing.

Thanks, Morton

Edit: Thanks for the replies. Being new to AG let me try and rephrase my quandary: It seems the projective setting is the most convenient to study AG but if I want to study properties of affine varieties, how does one use results of projective varieties in the affine case? I know this sounds vague but it is a fundamental doubt I have.

Answer 1

I think the question raises a valid point.

A very fruitful approach to affine problems was initiated by Iitaka in the 70's which is as follows:

Suppose $V$ is an affine variety and $X$ is a projectivisation such that $D=X-V$ is a divisor with simple normal crossings (SNC). Look at the canonical divisor $K:=K_X$ and the divisor $L:=K+D$ on $X$. Just like, the now classical, theory of Kodaira and others of analysing the multicanonical systems $nK$ of a projective variety, Iitaka proposed to look at $n(K+D)$ to come up with the a kind of classification for the pair $(X,D)$ as one does for projective varieties. Of course the only complete success in classifying varieties until Iitaka's time was for curves and Surfaces (which is also available now for 3-folds), so he and others applied this idea for non-compact (in particular affine) surfaces. Below I shall talk only about surfaces since the appropriate theory for 3-folds has not yet been worked out (as far as I know) and the curve case is extremely well understood and presents no real difficulty, generally speaking.

Just like a Kodaira dimension for projective surfaces, we can define the logarithmic Kodaira dimension of non-compact surfaces which is by definition the rate of growth of $n(K+D)$ as $n$ varies over positive integers. This number, called $\bar\kappa$ can take values $-\infty,0,1,2$ (or upto the dimension of the variety in the general case). At this stage one proves a theorem that this number is independent of the compactification $X$ chosen, as long as $D$ is SNC. This gets the theory started and we get a perfect gadget for studying the non-compact (in particular affine) surfaces. The whole project follows Kodaira's classification philosophy that one should develop enough classification theorems for the various $\bar\kappa$ classes and therefore (ideally) answer "all" questions about the non-compact (or affine) varieties. So if you want to answer a question like "are two affine varieties $A,B$ isomorphic or not" then the first thing to look at is their log-Kodaira dimensions. If they turn out to be different then we are done. If they are same then we have to look more closely into that particular $\bar\kappa$ class and either apply the appropriate classification theorems available or formulate and prove one, to decide.

However, just like in the projective case, the general type surfaces are hardest to study and don't always admit any good structure like a fibration over a curve which might have helped in their systematic study. And, by and large, the greatest success story has been in the non general type cases where there is a detailed classification of projective surfaces. Similar difficulties are encountered in the affine case and the $\bar\kappa\leq{1}$ affine surfaces are amenable to detailed study. Of course, there are some strong results about general type surfaces also which are in spirit the same as in the case of surface geography problem.

To find out more about these things one may look at Iitaka's book(GTM,76) and Miyanishi's book.

Answer 2

The most useful thing here for your interests would probably be log geometry, from what I know. In one formulation (Matsuki "Introduction to the Mori Program" is my reference) you look at pairs $(X,D)$ where $X$ is a projective variety and $D$ is a divisor on $X$ with normal crossings. Then look at the category of these pairs called log pairs, and the geometry captures a lot of the geometry of $X\setminus D$.

Here's the book, check out chapter 2, hope it helps.

Answer 3

Here is a simple example. Someone asked me if the unit disc is an affine variety. the answer is no. To see why not, assume yes, and take the projective closure. then one gets a projective curve which is a compact 2 dimensional surface with some points identified, and which differs from the original surface by at most adding a finite number of points. But this is impossible. No compact surface can be reduced to a disc by removing a finite number of points, even topologically, except for removing one point from P^1. But that does not give the disc by Liouville's theorem.

A more significant and pervasive example is the fact that at every singular point of an affine variety, the tangent cone determines a projective variety. Thus projective geometry is the local aspect of affine geometry. Put another way, blowing up an affine variety, at a point say, introduces projective geometry into it as a picture of its infinitesimal structure.

One can sometimes use this trick to compute the degree of a proper map of affine or other varieties, by restricting to the behavior over the projective normal bundle of a single fiber. See e.g. Friedman - Smith (Inventiones, 67, (1982)), who compute the degree of the prym map by showing that a single projective fiber of the proper prym map is embedded in projective space by the derivative of the prym map acting on the normal bundle to the fiber.

Answer 4

Different affine varieties can have a same projective closure. For example, look at $P^1(C)$, then $C$ is an affine open subset, and $C-{0}$ is also an affine open subset. But in any case, if two affine varieties have a same projective closure, they are birational.